Mathematics Rhyme and Reason

194 Pages·2018·25.451 MB·English

by  Currie, Mel

Mathematical Circles Library
Mathematics:
Melvin Currie
Rhyme and Reason
William Fleissner
Kenneth Kunen
Dana Scott
Alonzo Church
Oswald Veblen
E. H. Moore
H. A. Newton
Michel Chasles
Siméon Denis Poisson
Pierre-Simon Laplace Joseph Louis Lagrange
Leonhard Euler
Johann Bernoulli
Mel Currie
Jacob Bernoulli
Mathematics:
Rhyme and Reason

Mathematical Circles Library
Mathematics:
Rhyme and Reason
Mel Currie
Berkeley, California
Advisory Board for the MSRI/Mathematical Circles Library
TituAndreescu TatianaShubin(Chair)
DavidAuckly ZvezdelinaStankova
H´el`eneBarcelo JamesTanton
ZumingFeng RaviVakil
TonyGardiner DianaWhite
NikolajN.Konstantinov IvanYashchenko
AndyLiu PaulZeitz
AlexanderShen JoshuaZucker
SeriesEditor: MaiaAverett,MillsCollege.
This volume is published withthe generous support of the Simons
Foundation and Tom Leighton and Bonnie Berger Leighton.
2010 Mathematics Subject Classification. Primary 00A09, 97A80.
For additional informationand updates on this book, visit
www.ams.org/bookpages/mcl-22
Library of Congress Cataloging-in-Publication Data
Names: Currie,Mel(MelvinRobert),1948-author.
Title: Mathematics: rhymeandreason/MelCurrie.
Description: Berkeley,California: MSRIMathematicalSciencesResearchInstitute;Providence,
Rhode Island : American Mathematical Society, [2018] | Series: MSRI mathematical circles
library;22|Includesbibliographicalreferencesandindex.
Identifiers: LCCN2018028165|ISBN9781470447960(alk. paper)
Subjects: LCSH:Mathematics–Studyandteaching(Higher)|Mathematics–Humor. |AMS:Gen-
eral–Generalandmiscellaneousspecifictopics–Popularizationofmathematics. msc|Math-
ematics education – General, mathematics and education – Popularization of mathematics.
msc
Classification: LCCQA11.2.C872018|DDC510.71/2–dc23
LCrecordavailableathttps://lccn.loc.gov/2018028165
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Dedicated to the memory of Angela E. Grant and Rudy L. Horne
v

Contents
Preface ix
Chapter 1. The Riddle 1
Chapter 2. Primes 3
Chapter 3. Some Geometry 9
Chapter 4. Mysterious Pattern 17
Chapter 5. Some Things Add Up. Some Don’t. 21
Chapter 6. A Tangential Remark 29
Chapter 7. Plus or Minus 35
Chapter 8. Making the Optimal Choice 41
Chapter 9. Impossibilities 47
Chapter 10. Magnitudes of Infinity 57
Chapter 11. The Inevitable (Sperner’s Lemma—The Brouwer
Fixed-Point Theorem) 67
Chapter 12. Consider the Sequence (Fibonacci and Golden Ratio) 81
Chapter 13. What Are the Chances? 91
Chapter 14. The Euler Line 99
Chapter 15. The Dissertation 109
Chapter 16. The Next Prime Number Is? (Gandhi’s Formula) 113
Chapter 17. Bulgarian Solitaire 119
Chapter 18. Which is Bigger? (ab versus ba) 131
Chapter 19. Fascinating 135
vii
viii CONTENTS
Chapter 20. From the Sublime to the Ridiculous 141
Chapter 21. A Few More Words 147
Photos and Pictures 151
Appendix A. Notation, etc. 161
Appendix B. Mysterious 167
Appendix C. Impossibilities 171
Appendix D. Magnitudes 173
Appendix E. Fascinating 175
Preface
When I arrived on the Yale campus over fifty years ago, I was two months
shy of my eighteenth birthday. I had never held a conversation with a math-
ematician and knew little to nothing about the culture of the mathematics
community. But, somehow, I believed that I wanted to be a mathematician.
Any savvy observer would have certainly concluded that my prospects were
grim. Nonetheless, by hook and by crook, I eventually found a path.
I have fashioned a manuscript that I wishhad been available to mewhen
Iwasmakingmywaythroughhighschoolanddreamingofbecomingamath-
ematician. Of course, I am trusting that such kids still exist. I suspect that
many adults who feel that they missed the mathematics boat, and regret
it, will find that this book resonates with them. I include what I deem to
be mathematical gems and intersperse stories about mathematicians. The
working title of this book was “Nursery Rhymes of Mathematics” because I
hoped that the book would go a long way in establishing a cultural founda-
tion in mathematics; an introduction to a perspective that most people are
currentlynotexposedtosufficientlyearlyintheirlives. Grapplingwithmany
of the gems requires various levels of sophistication, but the tools needed are
notbeyond what manycollege-bound students willhave mastered by twelfth
grade. Mostofthe"nurseryrhymes"requireonlybasichighschoolalgebraor
geometry. Thebook isakindofsampler. Thechapters arearranged roughly
in the order of difficulty and can be read independently of each other. Some
students might take years to peruse all of them, with or without the support
of a teacher.
I did not avoid humor in writing Mathematics: Rhyme and Reason and
I was certainly willing to indulge my sentimental side in certain passages,
since this book is also serving as something like a memoir. Topics covered
include:
• The Infinitude of Primes
• Infinite Series
• No Four Points in the Plane with Pairwise Distances All Odd
• Magnitudes of Infinity/Existence of Transcendental Numbers
• Sperner’s Lemma/Brouwer Fixed Point Theorem
• Binet’s Formula
ix